@jfhm_bours: "I was wondering, in the light of Kurt G�dels findings, if it would be
possible to use another way of reasoning
than on the basis of axiomas in order to avoid the incompleteness-problems
inherent to aximatic reasoning.
I am very curious to know the different opinions on this subject!
Jan Bours
luke82@example.net"
    @kovarik: "<quote>
s
Yes: truth by decree; in one system (which I remember only too
well), the ultimate test of validity was whether a statement
serves the working class. It worked, with the help of a strong
police force, for some time.
Another, less violent but sillier, is truth by ballot (theorem
proved by a majority of at least fifty per cent plus one vote).
Cheers, ZVK(Slavek)."
    @drm39: "<quote>
To the best of my knowledge, there is nothing else which has been 
explored on a formal basis. Axiomatic reasoning is fairly central to 
mathematics after all.
The main alternative I suppose is intuitive mathematics; you work from 
what 'looks true' to prove your results. In practice this is mostly 
equivalent to axiomatic mathematics though - Generally what you think 
looks true is going to lie in some axiomatic theory. Also, a lot of 
mathematicians work in intuitive mathematics while officially declaring 
themselves to be working in an axiomatic system, which is fine - it's 
rarely worth checking through a precise formal proof in the system, as 
long as you can convince yourself and others that you could turn it into 
a formal proof if you were so inclined to do so (which I find I rarely 
am, on the grounds that it would be really really boring :).
The major problems with this are as follows:
1. Two different mathematicians could disagree as to what 'looks true'. 
The most common examples of this are statements like the axiom of choice 
and the continuum hypothesis. In practice this tends not to be a problem 
most of the time, but it could in principle be.
2. Often what looks true doesn't make a great deal of sense when you 
think about it harder. For example you are presumably familiar with 
Russel's paradox? i.e. Consider the set { x : x is not an element of x 
}. So how do we know our intuition is consistent (this is of course not 
avoided entirely by axiomatic mathematics, but it minimises the problem)
3. You often end up with statements that you have no intuition about and 
can't be reduced to anything you do have a better intuition of. This 
typically just means you avoid the question, but that can be a problem 
if it's something you want to use. (Of course this can happen in an 
axiomatic theory as well, but at least there you have an actual notion 
of undecidability and the possibility of adding in your statement as an 
axiom).
All this aside, I would argue that there is no incompleteness problem 
inherent to axiomatic reasoning.
Before you start shouting, I'm not disputing the incompleteness 
theorems. They are nice, elegant and very interesting results.
It is, however, not a problem.
Consider first the second incompletness theorem: If a system is 
consistent, it cannot prove it's own consistency. I really can't see how 
this is a problem - if a system is inconsistent then it can prove it's 
own consistency; therefore a proof within a system of it's own 
consistency seems inherently worthless. You're always going to need to 
refer to some bigger system (e.g. intuitive reasoning)... and how do you 
know that's consistent? Consistency is an annoying problem to have worry 
about, but it's always going to be a problem regardless of what approach 
you take. So it's interesting, but not really terribly relevant.
The first incompleteness theorem is much more interesting: It says there 
are always undecidable results (assuming your theory is sufficiently 
interesting). To my mind this is far from a problem: It basically says 
that no matter how much you know, there's always more mathematics that 
you can discover. In an infinity of mathematics there's bound to be 
various interesting things we don't know about, so the first 
incompleteness theorem says nothing more than that there will always be 
new and interesting mathematics to explore, no matter how much we 
already know! This is hardly a problem...
David
(E-mail address spam-blocked in the obvious way)"
    @spamtrap: "In <email_address>,
on 12/04/2003
   at 10:45 AM, "Zdislav V. Kovarik" <email_address>
said:
<quote>
ITYM whether it served the party; the opinions of the working class
were not solicited. From each according to his abilities, to each
according to his party rank.
     Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolicited bulk E-mail will be subject to legal action.  I reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact me.  Do
not reply to <email_address>"
    @hjurjus: ""Jeffrey H" <email_address> wrote in message news:<email_address>
[snip]
Can you imagine math without Euclid?  It would be very hard
<quote>
Even more: mathematical reasoning is exact, and involves _steps_.
So, no matter what you do, these steps can be analyzed and then
formalized as an axiom system.
However, there are forms of reasoning that are not based on steps
or are not 'exact' enough to be so easily formalized.
(For example: everyday life reasoning). Attempts to formalize these
are sometimes done using fixpoint constructions, and other tricks.
You might be interested to Google for non-monotonic logic.
Especially the argumentation-theoretic approach (Phan Minh Dung,
Henry Prakken) and inheritance networks (John Horty).
Hope this brings you further.
Good luck and cheers,
Herman Jurjus"
    @spamtrap: "In <email_address>, on 12/19/2003
<quote>
Yes.
But very easy to think of axioms without Euclid.
What generalized notion did G�del require? And what was Euclid's
notion of infinity?
What definitions are those? Why would they be relevant to G�del's
incompleteness results?
How does one imagine an infinite geometrical set that is not an
infinite set?
Incorrect; I'm far from the only one to find it intuitive.
What gives you the idea that sets have anything to do with swapping?
Also, two sets with different members are *not* the same set.
How can the human imagination grasp an opera in a foreign language?
Perhaps your imagination is deficient; that doesn't mean that
everyone's imagination is.
But others, better educated, do think that it does.
And what way was that? None of them touched the heart of Mathematics
the way that, e.g., Galois, Gauss, G�del, Hilbert, Klein and Reimann
did.
     Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolicited bulk E-mail will be subject to legal action.  I reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact me.  Do
not reply to <email_address>"
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